Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory

This study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam’s depth and calculated by using classical rule of mixture and Mori–Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton’s principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson’s ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined

Effect of elastic support on the linear buckling response of quasi-isotropic cylindrical shells under axial compression

Cylindrical shells under compressive loading are highly sensitive to boundary conditions. Considering that these structures are connected by surrounding structural components with finite stiffness, an accurate evaluation of the effects of their boundary stiffness is crucial in their design. As such, this work investigates the effect of elastic boundary conditions on the linear buckling behaviour of cylindrical shells under compressive loading. To achieve this goal, a virtual testing investigation on the effect of translational and rotational constraints to the linear buckling response of a quasi-isotropic cylinder subjected to axial compression is performed. Subsequently, the effect of many kinds of constraints on linear buckling behaviour is discussed and interesting insights regarding a significant coupling effect between the radial and tangential translational constraints are given. Results obtained from virtual testing show that seven recurrent buckling mode shapes occur with seven corresponding similar linear buckling loads. Therefore, based on these similarities, seven groups of classical boundary conditions are introduced to classify all possible linear buckling behaviours exhibited by the cylinder under consideration. Finally, these findings can support the development of theoretical models for cascade, or flange, designs of multiple connecting cylinders

A strain-displacement mixed formulation based on the modified couple stress theory for the flexural behaviour of laminated beams

A novel strain-displacement variational formulation for the flexural behaviour of laminated composite beams is presented, which accurately predicts three-dimensional stresses, yet is computationally more efficient than 3D finite element models. A global third-order and layer-wise zigzag profile is assumed for the axial deformation field to account for the effect of both stress-channelling and stress localisation. The axial and couple stresses are evaluated from the displacement field, while the transverse shear and transverse normal stresses are computed by the interlaminar-continuous equilibrium conditions within the framework of the modified couple stress theory. Then, axial and transverse force equilibrium conditions are imposed via two Lagrange multipliers, which correspond to the axial and transverse displacements. Using this mixed variational approach, both displacements and strains are treated as unknown quantities, resulting in more functional freedom to minimise the total strain energy. The differential quadrature method is used to solve the resulting governing and boundary equations for simply-supported, clamped and cantilever laminated beams. The deflections and stresses from this variational formulation for simply supported beams agree well with those from a Hellinger-Reissner stress-displacement mixed model found in the literature and the 3D elasticity solution given by Pagano. These strain-displacement models also accurately predict the localised stresses near clamped and free boundaries, which is confirmed by the high-fidelity Abaqus models

Design considerations for composite cylindrical shells on elastic foundations subject to compression buckling

Elastic boundary conditions play an important role in the buckling analysis of cylinders under compressive loading. These structures are used widely in aerospace applications and are highly sensitive to geometrical, material, loading, and boundary imperfections. In fact, the presence of these imperfections can lead to catastrophic failure. In 1968, NASA reported relations for obtaining the Knockdown Factor (KDF) based on an empirical method that is valid for isotropic and orthotropic materials; however, these relations do not consider the effect of elastic boundaries that can lead to highly conservative values of KDF. In design practice, a universal KDF of 0.65 has been used for recent designs by NASA, which may not be applicable to new types of structural configuration with different loading and boundary conditions. Therefore, there is a need for robust design factors for future designs which reduce the dependency on testing during preliminary design phases and speeds up the product development process. The availability of up‐to‐date and different KDF expressions for different structural configurations would help engineers to design lighter structures with improved load carrying capacity and reliability. The main objective of this work is to identify the buckling load sensitivity of cylindrical shells due to their boundary conditions and develop KDF relations considering elastic boundaries. To achieve this goal, the effect of axial, radial and tangential support stiffness on a quasi‐isotropic cylinder under axial compression is investigated. A data‐driven design approach is used to develop new KDF empirical relations for a quasi‐isotropic cylinder on different elastic foundations. The accuracy of these relations is within 5% for any elastic foundation considered

A strain-displacement variational formulation for laminated composite beams based on the modified couple stress theory

1. Introduction Mixed energy methods, e.g. Hellinger-Reissner (HR) method, can include stresses and displacements as unknowns in the formulation and can be made to satisfy equilibrium conditions as well as minimise total energy. The HR method accurately captures the full 3D stress field of general beams and plates [1]. The differential quadrature method (DQM) can be viewed as a more accurate, computationally efficient version of the finite difference method. It solves differential equations directly and therefore solves pointwise equilibrium expressions and can be more accurate than conventional FE methods. The present study aims to: • develop strain-displacement (SD) mixed energy method for stress analysis of laminated beams. • include couple-stress in the SD formulation to accurately predict 3D stresses of highly heterogeneous laminates. • implement DQM method to solve the governing equations subject to different boundary conditions

A mixed inverse differential quadrature method for static analysis of constant- and variable-stiffness laminated beams based on Hellinger-Reissner mixed variational formulation

Increasing applications of laminated composite structures necessitate the development of equivalent single layer (ESL) models that can achieve similar accuracy but are more computationally efficient than 3D or layer-wise models. Most ESL displacement-based models do not guarantee interfacial continuity of shear stresses within laminates. A possible remedy is the enforcement of interlaminar equilibrium in variational formulations, for example, in the framework of the Hellinger-Reissner variational principle, leading to a mixed force/displacement model. In this paper, the governing equations for bending and stretching of laminated beams, comprising only seven stress resultants and two displacement functionals, are obtained using global fifth-order and a local linear zigzag kinematics. As a strong-form solution technique, the differential quadrature method (DQM) is an efficient tool which can provide excellent convergence with relatively few number of grid points. However, in dealing with high-order differential equations, the conventional DQM can incur considerable errors due to the nature of numerical differentiation. Therefore, a mixed inverse differential quadrature method (iDQM) is proposed herein to solve the governing fourth-order differential equations for bending and stretching of laminated beams. This approach involves approximating the first derivatives of functional unknowns, thereby reducing the order of differentiation being performed. Using a non-uniform Chebychev-Gauss-Lobatto grid point profile, numerical results show that the accuracy of stress predictions is improved by using iDQM compared to DQM. In addition, the Cauchy’s equilibrium condition is satisfied more accurately by iDQM, especially in the vicinity of boundarie

Inverse differential quadrature method: mathematical formulation and error analysis

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions